Spring does provide the Task executor abstraction layer that allows you to run tasks in the background. You can configure them to run in threadpool. It also allows you to integrate with quartz which allows you to run your tasks periodically.
What you use, and how you use it depends on what you need to do. If you can elaborate, we can help you point in the right direction.
You need to provide more details. Is the application being tested built using Spring, or are you planning to leverage Spring to do your testing? Are you concerned how your Spring managed components will behave in a multi threaded test environment? Are you trying to test concurrency, or are your just trying to speed up your testing by parallelizing it?
If all you want to run your tests in parallel, you really don;t need Spring. You just divide up your tests into Suites, and make sure each suite can be tested independently. Then all you do is kick off multiple JUnit tests, one for each suite. Boom, parallelized!